Lifting vector fields to the rth order frame bundle

J. Kurek; W. M. Mikulski

J. Kurek ; W. M. Mikulski

Colloquium Mathematicae, Tome 111 (2008), p. 51-58 / Harvested from The Polish Digital Mathematics Library

Résumé

We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r} M = i n v J ₀^{r} (ℝ^{m}, M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r} M$ . In both cases we deduce that the spaces of all operators in question form free $(m (C_{r}^{m + r} - 1) + 1)$ -dimensional modules over algebras of all smooth maps $J ₀^{r - 1} T ̃ ℝ^{m} \to ℝ$ and $J ₀^{r - 1} T ℝ^{m} \to ℝ$ respectively, where $C ⁿ_{k} = n! / (n - k)! k!$ . We explicitly construct bases of these modules. In particular, we find that the vector space over ℝ of all natural linear operators lifting vector fields X on m-manifolds M to vector fields on $L^{r} M$ is $(m ² C_{r - 1}^{m + r - 1} (C_{r}^{m + r} - 1) + 1)$ -dimensional.

Publié le : 2008-01-01

Zbl 1132.58002

EUDML-ID : urn:eudml:doc:284029

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-5,
     author = {J. Kurek and W. M. Mikulski},
     title = {Lifting vector fields to the rth order frame bundle},
     journal = {Colloquium Mathematicae},
     volume = {111},
     year = {2008},
     pages = {51-58},
     zbl = {1132.58002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-5}
}

J. Kurek; W. M. Mikulski. Lifting vector fields to the rth order frame bundle. Colloquium Mathematicae, Tome 111 (2008) pp. 51-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-5/